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In this thesis which is a compendium of seven papers, we explore the behaviour of solutions of various semilinear elliptic equations and systems on both bounded and unbounded domains of dimension N.

On unbounded domains which is the main part of our work, our motivation is a celebrated conjecture of De Giorgi (1978) stating that bounded and monotone solutions of the Allen-Cahn equation, up to dimension 8, must be one-dimensional. This conjecture is known to be true for N=<3 and with extra (natural) assumptions for 4=<N=<8.

Focusing on system of equations, we state a counterpart of the above conjecture for gradient systems introducing the concept of monotonicity for systems. Then, we prove this conjecture for dimensions up to three following ideas given for the scalar case but unfortunately for higher dimensions we are not able to give any (even partial) results. On the other hand, replacing the Laplacian operator by the divergence form operator for the Allen-Cahn equation, we ask under what conditions solutions of this new equation would be m-dimensional. This leads us to define the concept of “m-Liouville theorem” for PDEs. We say a PDE satisfies m-Liouville theorem for 0<=m<N if all solutions of the PDE are at most m-dimensional. The motivation to this definition is the Liouville theorem (or 0-Liouville theorem) that we have seen in elementary analysis stating that bounded harmonic functions on the whole space must be constant (0-dimensional). We present various 2- and higher- Liouville theorems, however, we are not sure whether or not any of these results are optimal. 0-Liouville theorem is at the heart of our work and this thesis includes various 0-Liouville theorems for the Henon-Lane-Emden system, Lane-Emden equation, Gelfand equation and gradient systems.

On bounded domains, following ideas observed for unbounded domains, we present regularity of solutions for gradient and twisted-gradient systems. The novelty here is a stability inequality that gives us the chance to adjust the known techniques and ideas to systems.

Abstract

A highlight in the study of quantum physics was the work of Knizhnik, Polyakov and Zamolodchikov (1988), in which they proposed a relation (KPZ relation) between the scaling dimension of a statistical physics
model in Euclidean geometry and its counterpart in the random geometry. Recently, Duplantier and Sheffield used the 2D Gaussian free field to construct the Liouville quantum gravity measure on a planar domain, and
gave the first mathematically rigorous formulation and proof of the KPZ relation in that setting. We have applied a similar approach to generalize part of their results to R^4 (as well as to R^(2n) for
n>=2). To be specific, we construct a random Borel measure on R^4 which formally has the density (with respect to the Lebesgue measure) given by the exponential of an instance of the 4D Gaussian free field.
We also establish the KPZ relation corresponding to this random measure. This is joint work with Dmitry Jakobson.

Abstract

At York, we have found first-year science majors coming to us from the Ontario high school system in general rather poorly prepared for first-year university mathematics. The result is very high drop-plus-fail rates in our first-year math courses and a resulting high attrition rate in the early years of our degree programs. A major source of the problem appears to be the widespread use in the schools of an approach heavily emphasizing the memorization of solution problem templates, an approach which leaves a majority of our incoming science majors with deficiencies in very basic algebra, trigonometry, and, even more problematic, their intuitive understanding of the basic operations of arithmetic. In this discussion, I will outline an approach I have developed involving 4-day, 4-hour-per-day intensive remediation sessions focused on changing the way such students approach mathematics. The program was begun in 2005 and significantly expanded in 2009, now handling between 15 and 20% of the incoming class each year. I will present statistics outlining the significant impact we have seen on student performance. The aim is to keep the presentation very informal, leaving lots of time for discussion, feedback and suggestions on possibilities for further improving the initiative.

Abstract

I will discuss uniqueness and Monge solution results for multi-marginal optimal transportation problems with a certain class of cost functions; this class arises naturally in multi-agent matching problems in economics. This result generalizes a seminal result of Gangbo and \'Swi\c{e}ch on multi-marginal problems. I will also discuss some related observations about multi-marginal optimal transport on Riemannian manifolds.

Abstract

Wigner stated the general hypothesis that the distribution of eigenvalue spacings of large complicated quantum systems is universal in the sense that it depends only on the symmetry class of the physical system but not on other detailed structures. The simplest case for this hypothesis concerns large but ﬁnite dimensional matrices. Spectacular progress was done in the past two decades to prove universality of random matrices with orthogonal, unitary or symplectic invariance. These models correspond to log-gases with respective inverse temperature 1, 2 or 4.

I will first review the many occurrences of these statistics from the random matrices universality class. I will then report on a joint work with L. Erd{\H o}s and H.-T. Yau, which yields universality for the log-gases at arbitrary temperature at the microscopic scale.

Abstract

Fluctuations of random matrix theory type have been known to occur in analytic number theory since Montgomery's calculation of the pair correlation of the zeta zeros, in the microscopic regime. At the mesoscopic scale, the analogy holds, through a limiting Gaussian field, which present an ultrametric structure similar to log-gases. In particular we will consider an analogue of the strong Szeg{\H o} theorem for L-functions.